Caputo-Fabrizio Fractional Derivative Research Articles

Study of fractional forced KdV equation with Caputo–Fabrizio and Atangana–Baleanu–Caputo differential operators

It is essential for mathematicians, physicists, and engineers to construct fractional mathematical models for specific phenomena and develop numerical or analytical solutions for these models. In this work, we implement the natural decomposition approach with nonsingular kernel derivatives to investigate the solution of nonlinear fractional forced Korteweg–de Vries (FF-KdV) equation. We first investigate the FF-KdV equation under the Caputo–Fabrizio fractional derivative. The similar equations are then examined using the Atangana–Baleanu derivative. This approach combines the decomposition method with the Natural transform method. The series solution of the suggested equations is thus obtained using the natural transform. The key benefit of this novel approximate-analytical approach is that it may provide an analytical solution for the FF-KdV problem in the form of convergent series with simple computations. For each equation, three unique situations are chosen to demonstrate and test the viability of the proposed method. To guarantee the competence and dependability of the proposed method, the nature for various values of the Froude number Fr have been provided. The present approach is also used to calculate solutions at various fractional orders. The approximate series solution’s behavior for various fractional orders has been graphically displayed. The outcomes demonstrate that the methodology is simple to use and reliable when applied to numerous fractional differential equations.

A fractional dual-phase-lag thermoelastic model for a solid half-space with changing thermophysical properties involving two-temperature and non-singular kernels

<abstract> <p>The thermal and mechanical properties of materials show differences depending on the temperature change, which necessitates consideration of the dependence of the properties of these materials on this change in the analysis of thermal stress and deformation of the material. As a result, in the present work, a mathematical framework for thermal conductivity was formulated to describe the behavior of non-simple elastic materials whose properties depend on temperature changes. This derived model includes generalized fractional differential operators with non-singular kernels and two-stage delay operators. The fractional derivative operators under consideration include both the Caputo-Fabrizio fractional derivative and the Atangana-Baleanu fractional derivative, in addition to the traditional fractional operator. Not only that, but the system of governing equations includes the concept of two temperatures. Based on the proposed model, the thermodynamic response of an unlimited, constrained thermoelastic medium subjected to laser pulses was considered. It was taken into account that the thermal elastic properties of the medium, such as the conductivity coefficient and specific heat, depend on the temperature. The governing equations of the problem were formulated and then solved using the Laplace transform method, followed by the numerical inverse. By presenting the numerical results in graphical form, a detailed analysis and discussion of the effects of fractional factors and the dependence of properties on temperature are presented. The results indicate that the fractional order coefficient, discrepancy index, and temperature-dependent properties significantly affect the behavior fluctuations of all physical domains under consideration.</p> </abstract>

Solution of time-fractional gas dynamics equation using Elzaki decomposition method with Caputo-Fabrizio fractional derivative.

In this article, Elzaki decomposition method (EDM) has been applied to approximate the analytical solution of the time-fractional gas-dynamics equation. The time-fractional derivative is used in the Caputo-Fabrizio sense. The proposed method is implemented on homogenous and non-homogenous cases of the time-fractional gas-dynamics equation. A comparison between the exact and approximate solutions is also provided to show the validity and accuracy of the technique. A graphical representation of all the retrieved solutions is shown for different values of the fractional parameter. The time development of all solutions is also represented in 2D graphs. The obtained results may help understand the physical systems governed by the gas-dynamics equation.

A new study on the Newell-Whitehead-Segel equation with Caputo-Fabrizio fractional derivative

<p>In this research, we propose a new numerical method that combines with the Caputo-Fabrizio Elzaki transform and the q-homotopy analysis transform method. This work aims to analyze the Caputo-Fabrizio fractional Newell-Whitehead-Segel (NWS) equation utilizing the Caputo-Fabrizio q-Elzaki homotopy analysis transform method. The Newell-Whitehead-Segel equation is a partial differential equation employed for modeling the dynamics of reaction-diffusion systems, specifically in the realm of pattern generation in biological and chemical systems. A convergence analysis of the proposed method was performed. Two-dimensional and three-dimensional graphs of the solutions have been drawn with the Maple software. It is seen that the resulting proposed method is more powerful and effective than the Aboodh transform homotopy perturbation method and conformable Laplace decomposition method in the results.</p>

Analysis of a nonlinear problem involving discrete and proportional delay with application to Houseflies model

<abstract><p>This manuscript established a comprehensive analysis of a general class of fractional order delay differential equations with Caputo-Fabrizio fractional derivative (CFFD). Functional analysis was used to examine the existence and uniqueness of the suggested class and to generate sufficient requirements for Ulam-Hyers (UH) type stability. Further, a numerical method based on Lagrange interpolation is used to compute approximate solution. Then, some applications in physical dynamics including a houseflies model and a Cauchy type problem were discussed to illustrate the established analysis with graphical illustrations.</p></abstract>

A mathematical model for simulating the spread of infectious disease using the Caputo-Fabrizio fractional-order operator

<p>We examined intraspecific infectious rivalry in a dynamic contagious disease model. A non-linear dynamic model that considers multiple individual categories was used to study the transmission of infectious diseases. The combined effect of parameter sensitivities on the model was simulated using system sensitivities. To investigate the dynamic behavior and complexity of the model, the Caputo-Fabrizio (C-F) fractional derivative was utilized. The behavior of the proposed model around the parameters was examined using sensitivity analysis, and fractional solutions included more information than the classical model. Fixed point theory was used to analyze the existence and uniqueness of the solution. The Ulam-Hyers (U-H) criterion was used to examine the stability of the system. A numerical approach based on the C-F fractional operator was utilized to improve comprehension and treatment of the infectious disease model. A more precise and valuable technique for solving the infectious disease model was used in MATLAB numerical simulations to demonstrate. Time series and phase diagrams with different orders and parameters were generated. We aimed to expedite patient recovery while reducing the frequency of disease transmission in the community.</p>

A mathematical fractional model of waves on Shallow water surfaces: The Korteweg-de Vries equation

<abstract><p>The homotopy perturbation transform method was examined in the present research to address the nonlinear time-fractional Korteweg-de Vries equations using a nonsingular kernel fractional derivative that Caputo-Fabrizio recently developed. We devoted our research to the nonlinear time-fractional Korteweg-de Vries equation and certain associated phenomena because of some physical applications of this equation. The results are significant and necessary for illuminating a range of physical processes. This paper considered an innovative method and fractional operator in this context to obtain satisfactory approximations to the provided issues. To solve nonlinear time-fractional Korteweg-de Vries equations, we first considered the Yang transform of the Caputo-Fabrizio fractional derivative. In order to confirm the applicability and efficacy of the provided method, we took into consideration two cases of the nonlinear time-fractional Korteweg-de Vries equation. He's polynomials were useful in order to manage nonlinear terms. In this method, the outcome was calculated as a convergent series, and it was demonstrated that the homotopy perturbation transform method solutions converge to the exact solutions. The main benefit of the suggested method was that it offered solutions with a high degree of precision while requiring minimal computation. Graphs were also used to illustrate the series solution for a certain non-integer orders. Finally, a comparison of both examples outcomes were examined using diagrams and numerical data. These graphs showed how the approximated solution's graph and the precise solution's graph eventually converged as the non-integer order gets closer to integer order. When $ \varsigma = 1 $, several numerical comparisons were conducted with the exact solutions. The numerical simulation was offered to illustrate the efficiency and reliability of the proposed approach. In addition, the behavior of the provided solutions was explained using a number of fractional orders. The theoretical analysis matched with the findings obtained using the current technique, and the suggested technique can be extended to tackle many higher-order nonlinear dynamics problems.</p></abstract>

Implicit Caputo-Fabrizio fractional differential equations with delay

"This article deals with some existence and uniqueness results for several classes of implicit fractional differential equations with delay. Our results are based on some fixed-point theorems. To illustrate our results, we present some examples in the last section. Keywords: Caputo-Fabrizio fractional order derivative, implicit, delay, fixed point."

Caputo-Fabrizio type fractional differential equations with non-instantaneous impulses: Existence and stability results

In the current article, we establish the existence and uniqueness of solutions to the initial value problem for nonlinear implicit fractional differential equations with non-instantaneous impulses, including the Caputo-Fabrizio fractional derivative. This derivative encapsulates non-local and memory-dependent effects in the system's behavior. The problem's complexity arises from its implicit nature, where the relationships between variables are not directly specified. This inherent intricacy is further compounded by the introduction of non-instantaneous impulses, signifying abrupt, intermittent changes within the system at specific junctures, which in turn trigger sudden transitions. At its core, the initial value problem is concerned with deciphering a system's behavior based on its initial conditions, necessitating a solution that adheres to the given equation while considering the presence of non-instantaneous impulses. To derive the existence results, the study leverages established mathematical tools, specifically, the standard Banach's fixed point theorem and Darbo's fixed point theorem connected to Kuratowski's measure of noncompactness (KMN). Furthermore, the Hyers-Ulam stability (HU-stability) of the given problem is discussed, which, in turn, enhances the utility and reliability of these solutions in practical applications. In order to demonstrate the applicability of the discovered results for various values of δ, we construct some examples towards the end. The practical implications of these results extend to a wide range of fields, offering improved modeling, control, and understanding of complex systems.

Fractional Nadeem trigonometric non-Newtonian (NTNN) fluid model based on Caputo-Fabrizio fractional derivative with heated boundaries

The fractional operator of Caputo-Fabrizio has significant advantages in various physical flow problems due to the implementations in manufacturing and engineering fields such as viscoelastic damping in polymer, image processing, wave propagation, and dielectric polymerization. The current study has the main objective of implementation of Caputo-Fabrizio fractional derivative on the flow phenomenon and heat transfer mechanism of trigonometric non-Newtonian fluid. The time-dependent flow mechanism is assumed to be developed through a vertical infinite plate. The thermal radiation’s effects are incorporated into the analysis of heat transfer. With the help of mathematical formulations, the physical flow system is expressed. The governing equations of the flow system acquire the dimensionless form through the involvement of the dimensionless variables. The application of Caputo-Fabrizio derivative is implemented to achieve the fractional model of the dimensionless system. An exact solution of the fractional-based dimensionless system of the equations is acquired through the technique of the Laplace transform. Physical interpretation of temperature and velocity distributions relative to the pertinent parameters is visualized via graphs. The current study concludes that the velocity distributions exhibit an accelerating nature corresponding to the increasing order of the fractional operator. Moreover, the graphical results are more significant corresponding to the greater time period.

The differential transform of the Caputo-Fabrizio fractional derivative

The differential transform of the Caputo-Fabrizio fractional derivative

Comparative Analysis and Definitions of Fractional Derivatives

Fractional Calculus (FC) has emerged as a valuable tool in various fields. This study explores the historical development of (FC) and examines prominent definitions regarding Fractional Derivatives (FD), such as the Riemann-Liouville, Grunwald-Letnikov, Caputo Fractional Derivative, Katugampula derivatives, Caputo Fractional Derivative, Caputo-Fabrizio Fractional Derivative and as well as Atangana-Baleanu Fractional Derivative. It critically evaluates their strengths, weaknesses and implications on (FD) equations. The findings contribute to establishing a clearer understanding of Fractional Derivatives (FD) and guiding their appropriate use in practical scenarios. It investigates the impact of different definitions on the properties and behaviors of (FD), providing valuable insights for researchers.

Exploring the couple stress flow of sodium alginate-based casson tetra hybrid nanofluid between inclined parallel microplates using Fourier analysis and Fick's laws

Compared to a regular fluid like water, the Couple Stress Fluid (CSF) has an additional material constant that manages both couple stress and lubricant viscosity. However, because this material constant creates a fourth-order derivative term in the momentum equation, there has been little research done on this type of fluid, even for traditional fluid problems. In this article, we analyze the flow of a tetra hybrid nanofluid consisting of Casson (sodium alginate-based) couple stress between two parallel microplates that are inclined infinitely. Fourier and Fick's laws are used to investigate the flow, where one plate remains stationary and the other oscillates at a constant speed. The flow is driven by the buoyant force generated by heat transfer and free convection, which results in the uniform dispersion of all spherical dust particles throughout the fluid. The system of partial differential equations (PDEs) is expanded using the Caputo-Fabrizio (CF) fractional derivative to simulate fluid flow. The problem is tackled by using finite sine Fourier and Laplace transforms to obtain closed-form solutions for temperature and velocity profiles. CF time fractional derivatives are used to provide CSF (Couette flow with suction or injection) results for various stimulating fluid parameters and examine their impact on the system. Graphical representations of the CSF, dust particle, and temperature profiles are presented, and skin friction and Nusselt number are calculated using Mathcad-15. It is found that an increase in the couple stress parameter leads to a retardation of both velocity profiles.

Effect of Newtonian heating on double convection flow fractional differential type fluid

The influence of Newtonian heating (NH) is studied on heat-mass transfer problem of fractional differential type fluid. The proposed problem is modeled by using Caputo-Fabrizio fractional order derivative. A steady magnetic field of uniform strength is enforced in a perpendicular direction of flow. The set of governing equations in dimensional form accompanied by suitable initial and boundary conditions are changed into non-dimensional form by making use of some non-dimensional variables. The resulting problem is then converted into a model employing fractional differential operator by using the new Caputo-Fabrizio fractional differential operator. The analytical exact solution has been obtained by making use of Laplace transform technique. The numerical results are computed via MATHCAD software and displayed in various plots for different physical parameters and discussed.

MHD blood flow effects of Casson fluid with Caputo-Fabrizio fractional derivatives through an inclined blood vessels with thermal radiation

This study investigates a fractional-order time derivative model of non-Newtonian magnetic blood flow in the presence of thermal radiation and body acceleration through an inclined artery. The blood flow is formulated using the Casson fluid model under the control of a uniformly distributed magnetic field and an oscillating pressure gradient. Caputo-Fabrizio's fractional derivative mathematical model was used, along with Laplace transform and the finite Hankel transform technique. Analytical expressions were obtained for the velocity of blood flow, magnetic particle distribution, and temperature profile. These distributions are presented graphically using Mathcad software. The results show that the velocity increases with the time, Reynolds number and Casson fluid parameters, and diminishes when Hartmann number increases. Moreover, fractional parameters, radiation values, and metabolic heat source play an essential role in controlling the blood temperature. More precisely, these results are beneficial for the diagnosis and treatment of certain medical issues.

Existence Theorems for Hybrid Fractional Differential Equations with ψ -Weighted Caputo–Fabrizio Derivatives

In this study, two classes of hybrid boundary value problems involving ψ -weighted Caputo–Fabrizio fractional derivatives are considered. Based on the properties of the given operator, we construct the hybrid fractional integral equations corresponding to the hybrid fractional differential equations. Then, we establish and extend the existence theory for given problems in the class of continuous functions by Dhage’s fixed point theory. Furthermore, as special cases, we offer further analogous and comparable conclusions. Finally, we give two examples as applications to illustrate and validate the results.

Stabilization and asymptotic stability of the Caputo–Fabrizio fractional-order linear and semilinear evolution equations

We discuss the asymptotic stability of the fractional-order linear and semilinear evolution equations involving a Caputo–Fabrizio fractional derivative for fractional-order α∈(0,1) with a non-singular kernel. First, by using the concepts of an equilibrium point, it is proved that an autonomous Caputo–Fabrizio system admits only a constant solution and a new concept of a global solution for the Caputo–Fabrizio system is introduced. Laplace transform and Grönwall inequality are used to derive the local and global asymptotic stability conditions. By constructing a suitable linear feedback control and using our main results, we stabilize the Caputo–Fabrizio fractional-order linear and semilinear evolution equations. At the end, by using the stabilization result, we stabilize a fractional-order chaotic system to support the obtained results.

Caputo-Fabrizio fractional order derivative stochastic resonance enhanced by ADOF and its application in fault diagnosis of wind turbine drivetrain

Fault diagnosis of wind turbine drivetrains is vital to maintain the reliability of wind turbines and stochastic resonance (SR) is regarded as a powerful method to amplify the fault-induced weak characteristics of vibration signals. However, without considering the high dependence between current and previous values of vibration signals, integer-order SR may not be effective enough to amplify the weak characteristics. Moreover, non-Gaussian noise originating from severe working condition generally reduce the efficiency of amplification. To address these issues, a Caputo-Fabrizio fractional order derivative (CF) SR improved by ascending density outlier factor (ADOF) for fault diagnosis of wind turbine drivetrains is proposed in this paper. First, ADOF is constructed to remove the large impulsive noise of vibration signals. Second, CF is discretized into the second-order SR model, then the weak characteristics of vibration signal are amplified by the established CF second-order SR model (CF-SR). The effectiveness is validated by a simulation and two experimentations based on two real vibration signals collected from the key components of wind turbines. Compared with traditional diagnosis methods such as deconvolution and Kurtogram, the proposed method is superior for fault diagnosis of wind turbines working in severe environments.

Heat transfer analysis of generalized second-grade fluid with exponential heating and thermal heat flux

The aim of present work is to apply the Caputo–Fabrizio fractional derivative in the constitutive equations of heat transfer. Natural convection flow of an unsteady second grade fluid over a vertical plate with exponential heating is discussed. The generalized Fourier law is substituted in temperature profile. A portion of the dimensionless factors are utilized to make the governing equations into dimensionless structures. The solutions for temperature and velocity profiles of Caputo–Fabrizio model are acquired through the Laplace transform method. These solutions are greatly affected through the variation of different dimensionless variables like Prandtl number, Grashof number, and second-grade fluid parameter. Finally, the influence of embedded parameters is shown by plotting graphs through Mathcad. From the graphical results it is concluded that, the temperature of the fluid decreases with the increasing values of the Prandtl number and Second grade fluid parameter and increases with the passage of time. The velocity of the fluid increases with increasing values of the Grashof number, second grade parameter and time while decreases with increasing values of fractional parameter and Prandtl number.

Irreversibility analysis in the ternary nanofluid flow through an inclined artery via Caputo-Fabrizio fractional derivatives

The current study deals with the fractional order flow of the Casson tri-nanofluid through an inclined artery in the existence of stenosis. Gold (Au), copper (Cu), and alumina (Al2O3) nanomaterials are added to the base fluid (blood) to form the tri-nanofluid. The flow is taken to be highly pulsatile. The mathematical formulation has been carried out by using differential forms of laws of conservation of mass, momentum, and energy. The dimensionless form of governing equations is simplified by applying the mild stenosis assumption. By using the Caputo-Fabriziotime-fractional derivative, the classical modeled problem is converted into its fractional equivalents. Utilizing both the Laplace and the Hankeltransforms, solutions to the velocity and temperature equations are found. The results of governing parameters on the fluid velocity, temperature, and entropy generation are deliberated graphically. The numerical outcomes for the velocity and temperature profiles and entropy generation are explored in graphical forms. The importance of considering the tri-nanofluid in the study is to enhance the heat transfer features by improving the thermal conductance of the fluid. The velocity profile increases with enhancing values of the Casson fluid parameter β. The temperature profile decreases with rising values of the fractional parameter α. The Entropy generation increases with growing values Ha while the reverse behavior is observed for the Bejan Be number. The Bejan profile declines with increasing values of Ha.